This thesis has studied global Torelli problems for projective manifolds. In particular, we have focused on projective manifolds of Calabi-Yau type, which is a generalization of Calabi-Yau manifolds.

In collaboration with F. Guan and K. Liu, we have proven the injectivity of the period map on the Teichmuller space of polarized and marked Calabi-Yau type manifolds. Our approach has been to construct the holomorphic affine structure on the Teichmuller space and the Hodge metric completion of the Teichmuller space. As a corollary, we also prove that the Hodge metric completion space of the Teichmuller space for Calabi-Yau type manifolds is a domain of holomorphy and it admits a Kahler-Einstein metric.

More generally, we are interested in global Torelli problems for projective manifolds. We have been working on adopting our techniques to more general projective manifolds, such as Calabi--Yau manifolds, projective hypersurfaces, and projective hyperkahler manifolds.

As direct applications, we will prove some properties for the period map on the moduli space of Calabi-Yau type manifolds; we will also prove a general result about the period map to be biholomorphic from the Hodge metric completion space of the Teichmuller space of Calabi-Yau type manifolds to their period domains, and apply it to the cases of K3 surfaces and cubic fourfolds. Moreover, we will discuss some special cases when the period domain has the same dimension as the Teichmuller space and has a natural affine structure.

It is well known that reconstruction problems, as the

cross-disciplinary subject, have been studied in numerous contexts

including statistical physics, information theory and computational

biology. My major contributions to the this field are to figure out

the tightness of the Kesten-Stigum reconstruction bound for both the

2q-state symmetric model with triple mutation probabilities and

the asymmetric binary channel on trees. Furthermore, we determine

asymptotics for the reconstruction thresholds on regular trees of

large degree.

In this thesis, we consider geometric properties of vector bundles

arising from algebraic and Hermitian geometry.

On vector bundles in algebraic geometry, such as ample, nef and globally generated vector

bundles, we are able to construct positive Hermitian metrics in

different senses(e.g. Griffiths-positive, Nakano-positive and

dual-Nakano-positive) by $L^2$-method and deduce many new vanishing

theorems for them by analytic method instead of the Le Potier-Leray

spectral sequence method.

On Hermitian manifolds, we find that the second Ricci curvature

tensors of various metric connections are closely related to the

geometry of Hermitian manifolds. We can derive various vanishing

theorems for Hermitian manifolds and also for complex vector bundles

over Hermitian manifolds by their second Ricci curvature tensors. We

also introduce a natural geometric flow on Hermitian manifolds by

using the second Ricci curvature tensor.

In this thesis, we study the geometry of Teichmuller space of punctured Riemann surfaces.

We use L2 Hodge theory to describe the deformation theory for punctured Riemann surfaces,

in which we defined Weil-Petersson metric, Hodge metric and Kodaira-Spencer map. We

also give a new proof of Wolpert's curvature formula by computing the expansion of volume

form and the Kodaira-Spencer map. We use Wolpert's formula to estimate upper bound for

various curvature tensor. We construct an extension of pluricanonical form and compare it

to the expansion of the Kodaira-Spencer map under Hodge metric.